The control members which are included in a rolling mill influence the flatness of the strip in different ways. The screws of the rolling mill are used for setting the roll gap across the strip or for adjustment or intentional angular adjustment of the roll gap. Normally bending cylinders are provided, both for bending of the work rolls and for bending of intermediate rolls in a 6-high rolling mill. Normally, also so-called shifting devices are included for axial shifting of the rolls.
A condition for achieving the desired flatness of the rolled product is to have a more or less continuous access to a measure of actual flatness across the strip, that is, a flatness curve. With a known flatness curve, the rolling mill can be provided with a closed-loop flatness control. In a classical manner, the flatness curve obtained is compared with the desired flatness. The flatness errors which thereby arise are then used, in accordance with different models, for influencing the control members to minimize the flatness errors. Thus, the flatness control comprises several executing devices, which means a relatively extensive evaluation process to decide on the magnitude of the various actions by the control members which provide the best result.
A very suitable measurement device--which is often used in these applications--for determining the flatness curve of the rolled strip is the "STRESSOMETER", developed by Asea Brown Boveri AB, which has been available on the market since the middle of the 60's and which has been described in a large number of pamphlets and other publications. The measurement device is designed as a measuring roll, with approximately 50 measuring points across the strip, which in most cases can be placed between the mill stand and the wind-up reel without the use of deflector rolls. The measurement takes place with the aid of force transducers, based on the magnetoelastic principle, and primarily provides the stress distribution of the strip along the measuring roll. If the stress is greater than the buckling stress for the material, the sheet buckles when the strip is left free with no influence by any tensile force. The stress distribution is a flatness curve for the strip across the rolling direction. A more detailed description of the measurement principle is given, inter alia, in an article in IRON AND STEEL ENGINEER, April, 1991, pp. 34- 37, "Modern approach to flatness measurement and control in cold mill" by A. G. Carlstedt and O. Keijser. The article discloses that, because of the relatively extensive signal processing which is required to obtain the flatness curve, this will be updated at intervals of about 50 ms.
When rolling strip, it is important to check and to have the correct roll gap since small variations along the work rolls give a varying reduction of the thickness across the strip, which in turn leads to an inferior flatness curve. The task of the flatness control is thus to maintain an existing curve constant during the whole rolling operation.
As is clear, among other things from the above-mentioned article in IRON AND STEEL ENGINEER, a technique is often used which comprises modifying, with the aid of the bending cylinders, the shape of the work rolls to influence the flatness of the strip. As will have been clear, however, there are several other control possibilities which can be used to influence the flatness curve. A concept for flatness control, in which several control members can be activated, is also described in the article mentioned. The concept includes a model comprising an evaluation strategy for which control members are to be activated as well as processing of collected measured data to obtain, by means of the least squares method, control signals to the control devices and the regulators for the different control members. In the example shown, the flatness control comprises skewing, axial shifting, and bending of the work rolls but in the general case it may comprise additional control possibilities.
In principle, the least squares method entails a possibility, each time the flatness error is updated, that is, after each comparison between the actual flatness curve and the desired flatness curve, of obtaining the combination and extent of actions by the control devices which are needed for the flatness error to be as small as possible. However, this method presupposes that the stress distribution, which arises across the strip when the different control members are activated, is known. The stress distribution can either be calculated or measured with the aid of the measuring roll. Assuming, as in the example shown, that there are three control members, for example skewing with a stress distribution .phi..sub.s, bending with a stress distribution .phi..sub.B, and axial shifting with a stress distribution .phi..sub.F, it is possible, using the least squares method, to indicate for each updated flatness error the actions by the different control members determined by EQU f*.sub.1 =c.sub.S .multidot..phi..sub.S +c.sub.B .multidot..phi..sub.B +c.sub.F .multidot..phi..sub.F ( 1)
where c.sub.S, c.sub.B and c.sub.F are the input signals to the control devices and regulators of the control members, which signals are converted into roll gaps. It is obvious that these calculations require very large computer capacity.
The approximation problem in general form comprises finding, with the aid of a number of measured data f(x.sub.i) with i=1, 2, . . . m, a simple function f* by means of the least squares method which approximates f(x.sub.i) as good as possible. The further description of the least squares method is based on the designations used in Larobok i Numeriska Metoder ("Textbook of Numerical Methods") by P Pohl, G Eriksson and G Dahlquist, published by Liber tryck, Stockholm. It is assumed here that the simple function f* is to be a linear combination of pre-selected functions .phi..sub.1, . . . .phi..sub.n according to EQU f*.sub.n =c.sub.1.phi.1 +c.sub.2.phi.2 + . . . +c.sub.n.phi.n ( 2)
and the task of the least squares method is then to determine c.sub.1, c.sub.2 . . . c.sub.n such that the sum of the squares of the deviations between f(x.sub.1) and f* is minimized.
The matrix formulation of the least squares method means that the following matrices are formed ##EQU1## with A=m.multidot.n where m=the number of measuring points=the number of lines in A and
n=the number of basic functions .phi..sub.1, . . . .phi..sub.n =the number of columns in A, ##EQU2## where f.sub.1, f.sub.2, . . . f.sub.m are the measured data obtained.
According to the least squares method, the following relationship applies between the matrices for determining c.sub.1, . . . c.sub.n : EQU A.sup.T A.multidot.c=A.sup.T .multidot.f (3)
where A.sup.T is the transposed matrix A. Without going further into the details of the method, the determination according to the prior art entails a time-consuming arrangement of the quadratic matrix A.sup.T A for each flatness curve.
From the point of view of feedback control, it is now desired to set up the functions .phi..sub.i which correspond to the mechanical actuator actions, for example the bending action which gives a flatness response of the form .phi..sub.B and then determine the corresponding c.sub.B together with the corresponding functions for the other control members.
From the computational point of view, this entails a considerable problem. With a calculation time of 0.15 ms per multiplication, the calculation time of the matrix for 3 control members and 50 measured values for each flatness curve will be about 160 ms, which means that it is not possible to evaluate each flatness curve.
There are different ways of solving this problem, which, however, entail reduced accuracy in the flatness control. One method of solution is disclosed by EP 0 063 606, "System for controlling the shape of a strip". Here, orthogonal functions are used where the quadratic matrix only contains a diagonal line with terms different from zero. The demands imposed by the control for functions which correspond to the actions are then abandoned and other functions are relied upon, and some interlinking is performed afterwards. The greatest disadvantage of this method is the restriction to polynomials and sine functions and that a higher order has to be used to approximate the flatness error in a satisfactory way.
Another method is disclosed in GB 2 017 974 A "Automatic control of rolling". In this case, the solution principle is to restrict the evaluation to a straight line and a parabola, that is, as "a curve of the form ax.sup.2 +c", as is clear, for example, from page 3, column 1, line 7 thereof.